Norms from quadratic fields and their relation to noncommuting 2×2 matrices III. A link between the 4-rank of the ideal class groups in ℚ(√m) and in ℚ(√-m)

Abstract

This paper is concerned with the representation of an integral 2 x 2 matrix A as A = S₁S₂ with S_i = S'_i and integral and facts connected with it. In [6] the following was shown. If the characteristic polynomial of A is x²-m with m square free and ≡2 or 3(4) then a factorization of A as above is only possible if the ideal class in Z[√m[ associated with A is of order a factor of 4. If the ideal class is of order 4 then the S_i cannot be unimodular. Now it is shown that a factorization for an A with characteristic polynomial x²-m, m square free, leads to an ideal class in the narrow sense of order 4 in Z[√-m]. This is achieved by associating with the factorization an integral ternary form representing zero in a nontrivial way. The conditions for this to happen are known.

Additional details